Four Methods:Calculating the Radius when you know the DiameterCalculating the Radius when you know the CircumferenceCalculating the Radius When You Know the AreaCalculating the Radius When You Know the Coordinates of Three Points on the CircleCommunity Q&A
The radius of a circle is the distance from the center of the circle to its edge. The diameter of the circle is the distance all the way across it, and is equal to two times the radius. You will often be asked to calculate the radius of the circle based on other measurements. This article will teach you to calculate the radius of a circle when you know its diameter, circumference, and its area. It will then teach you a more advanced method of finding the center and radius of a when you know the coordinates of three points that lie on the circle.
Calculating the Radius when you know the Diameter
1Remember what a diameter is. The diameter of a circle is the length of the line drawn through the center of a circle that touches the circle at two points. The diameter is the longest line that can be drawn through a circle, and cuts the circle into two equal halves. It is also two times the length of the radius. The formula for the diameter is D = 2r, where “D” stands for diameter, and “r” stands for radius. The same formula in terms of r is r = D/2.
2Divide the diameter by 2 to find the radius. If you are given the diameter of the circle, simply divide it by 2 to find the radius.
- For example, if the diameter of a circle is 4, the radius is equal to 4/2, or 2.
Calculating the Radius when you know the Circumference
1Recall the formula for the circumference of a circle. The circumference of the circle is the distance around it. Another way to think of it is that the circumference is the length of the line you would get if you cut the circle open and stretched it out flat. The formula for the circumference of a circle is C = 2πr, where “r” is the radius, and π is the constant pi, or 3.14159... The formula for the radius in terms of the circumference is then r = C/2π. 
- It is usually OK to round pi off the hundredths place (3.14), but check with your teacher to find out how many places she would like you to use.
2Calculate the radius from the circumference. To calculate the radius based on the circumference, simply divide the circumference by 2π, or 6.28.
- For example, if the circumference of a circle is 15, the radius r = 15/2π, or 2.39.
Calculating the Radius When You Know the Area
1Recall the formula for the area of a circle. The area of a circle is defined as A = πr2. If we restate that formula in terms of r, it becomes: r = √(A/π) (“r equals the square root of the Area divided by pi”).
2Plug the area into the formula. For example, if the area of the circle is 21 in 2 ; when we plug that value into the formula, we get: r = √(21/π).
3Divide the area by π (3.14).
- 21 / 3.14 = 6.69.
4Use your calculator to find the square root of this number. The result will be the radius of your circle.
- In our example, √6.69 = 2.59, the radius of our circle.
Calculating the Radius When You Know the Coordinates of Three Points on the Circle
1Understand that three points can define a circle. Any three points on a coordinate plane will uniquely define a circle that touches all three points. The center of this circle may lie inside or outside of the circle, depending on how the points are arranged, and is called the “circumcenter” of the triangle. The radius of this circle is called its circumradius. It is possible to calculate this radius if you know the (x,y) coordinates of the three points in question.
- As an example, let’s say that three known points on our circle are defined as: P1 = (3,4), P2 = (6, 8), and P3 = (-1, 2).
2Use the distance formula to calculate the lengths of the three sides of the triangle, which we will label a, b, and c. The distance formula states that the distance between two points on a coordinate plane (x1, y1) and (x2, y2) is: distance = √(( x2 - x1)2 + (y2 - y1)2). Plug the coordinates into this formula to find lengths for each of the three sides of the triangle.
3Calculate the length of the first side a, which goes from point P1 to point P2. In our example, the coordinates of P1 are (3,4) and P2 is at (6,8), so the length of side a = √((6 - 3)2 + (8 - 4)2).
- a = √(32 + 42)
- a = √(9 + 16)
- a = √25
- a = 5
4Repeat the process to find the length of the second side b, which goes from point P2 to point P3. In our example, the coordinates of P2 are (6,8) and P3 is at (-1,2), so the length of side b is defined as: b =√((-1 - 6)2 + (2 - 8)2).
- b= √(-72 + -62)
- b = √(49 + 36)
- b = √85
- b = 9.23
5Repeat the process to find the length of the third side c, which goes from point P3 to point P1. The coordinates of P3 are (-1,2) and P1 is at (3,4), so the length of side c is defined as: c =√((3 - -1)2 + (4 - 2)2).
- c= √(42 + 22)
- c = √(16 + 4)
- c = √20
- c = 4.47
6Now enter the lengths into the formula for finding the circumradius, (abc)/(√(a + b + c)(b + c - a)(c + a - b)(a + b - c)). The answer will be the radius of our circle!
- For our triangle, a = 5, b = 9.23 and c = 4.47. So our formula for the radius will look like this: r = (5 * 9.23 * 4.47)/(√(5 + 4.47 + 9.23)(4.47 + 9.23 - 5)(9.23 + 5 - 4.47)(5 + 4.47 – 9.23)).
7First multiply the three lengths together to find the numerator of the fraction. Then update the formula.
- (a * b * c) = (5 * 9.23 * 4.47) = 206.29
- r = (206.29)/(√(5 + 4.47 + 9.23)(4.47 + 9.23 - 5)(9.23 + 5 - 4.47)(5 + 4.47 – 9.23))
8Add together the values in each set of the parentheses. Then plug them into the formula.
- (a + b + c) = (5 + 4.47 + 9.23) = 18.7
- (b + c - a) = (4.47 + 9.23 - 5) = 8.7
- (c + a - b) = (9.23 + 5 - 4.47) = 9.76
- (a + b - c) = (5 + 4.47 – 9.23) = 0.24
- r = (206.29)/(√(18.7)(8.7)(9.76)(0.24))
9Multiply the values in the denominator together.
- (18.7)(8.7)(9.76)(0.24) = 381.01
- r = 206.29/√381.01
10Calculate the square root of the product to find the denominator of the fraction.
- √381.01 = 19.51
- r = 206.29/19.52
11Now divide the numerator by the denominator to find the radius of the circle!
- r = 10.57
How do I find the measure of an arc without a radius?
- Figure out the ratio of the length of the arc to the circumference and set it equal to the ratio of the measure of the arc (shown with a variable) and the measure of the entire circle (360 degrees). For example: If the circumference of the circle is 4 and the length of the arc is 1, the proportion would be 4/1 = 360/x and x would equal 90.
How do I calculate the radius of a circle when no other values are known?
- Technically you can't "calculate" the radius in such a situation. However, it is possible, by construction, to locate the center of such a circle, and then, simply by physically measuring, determine the radius. To do the construction, draw any two chords and construct their perpendicular bisectors; their point of intersection is the center of the circle. Then draw in any radius and measure it with a ruler. Not technically a "calculation."
Sources and Citations
- ↑ https://www.mathsisfun.com/definitions/radius.html
- ↑ http://www.mathopenref.com/radius.html
- ↑ http://mathforum.org/dr.math/faq/faq.pi.html
- ↑ http://www.mathgoodies.com/lessons/vol2/circle_area.html
- ↑ https://www.khanacademy.org/math/geometry/triangle-properties/perpendicular_bisectors/v/area-circumradius-formula-proof
- ↑ http://www.mathopenref.com/trianglecircumcircle.html
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